User:Tohline/Appendix/Ramblings/T3Integrals

Integrals of Motion in T3 Coordinates

Motivated by the HNM82 derivation, in an accompanying chapter we have introduced a new T2 Coordinate System and have outlined a few of its properties. Here we offer a modest redefinition of the second radial coordinate in an effort to bring even more symmetry to the definition of the position vector, <math>\vec{x}</math>.

Definition

By defining the dimensionless angle,

<math> \Zeta \equiv \sinh^{-1}\biggl( \frac{qz}{\varpi} \biggr) , </math>

the two key "T3" coordinates will be written as,

<math> \lambda_1 </math>

<math>\equiv</math>

<math>\varpi \cosh\Zeta = ( \varpi^2 + q^2z^2 )^{1/2}</math>

and

<math> \lambda_2 </math>

<math>\equiv</math>

<math>\varpi [\sinh\Zeta ]^{1/(1-q^2)} = \biggl[\frac{\varpi^{q^2}}{qz}\biggr]^{1/(q^2-1)}</math>

Here are some relevant partial derivatives:

	<math> \frac{\partial}{\partial x} </math>	<math> \frac{\partial}{\partial y} </math>	<math> \frac{\partial}{\partial z} </math>
<math>\lambda_1</math>	<math> \frac{x}{\lambda_1} </math>	<math> \frac{y}{\lambda_1} </math>	<math> \frac{q^2}{\lambda_1} </math>
<math>\lambda_2</math>	<math> \frac{1}{(q^2-1)} \biggl[ \frac{\varpi^{q^2-1}}{\sinh\Zeta} \biggr]^{q^2/(q^2-1)} \biggl( \frac{q^3 z}{\varpi^{q^2+2}} \biggr) x </math> <math> =\frac{q^2}{(q^2-1)} \biggl[ \frac{\varpi^{q^2}}{qz} \biggr]^{1/(q^2-1)} \biggl( \frac{x}{\varpi^2} \biggr) </math>	<math> \frac{1}{(q^2-1)} \biggl[ \frac{\varpi^{q^2-1}}{\sinh\Zeta} \biggr]^{q^2/(q^2-1)} \biggl( \frac{q^3 z}{\varpi^{q^2+2}} \biggr) y </math> <math> =\frac{q^2}{(q^2-1)} \biggl[ \frac{\varpi^{q^2}}{qz} \biggr]^{1/(q^2-1)} \biggl( \frac{y}{\varpi^2} \biggr) </math>	<math> - \frac{1}{(q^2-1)} \biggl[ \frac{\varpi^{q^2-1}}{\sinh\zeta} \biggr]^{q^2/(q^2-1)} \frac{q}{\varpi^{q^2}} </math> <math> =- \frac{1}{(q^2-1)} \biggl[ \frac{\varpi^{q^2}}{qz} \biggr]^{1/(q^2-1)} \frac{1}{z} </math>
<math>\lambda_3</math>	<math> - \frac{y}{\varpi^{2}} </math>	<math> + \frac{x}{\varpi^{2}} </math>	<math> 0 </math>

The scale factors are,

<math>h_1^2</math>	<math>=</math>	<math> \biggl[ \biggl( \frac{\partial\lambda_1}{\partial x} \biggr)^2 + \biggl( \frac{\partial\lambda_1}{\partial y} \biggr)^2 + \biggl( \frac{\partial\lambda_1}{\partial z} \biggr)^2 \biggr]^{-1} </math>	<math>=</math>	<math> \lambda_1^2 \ell^2 </math>
<math>h_2^2</math>	<math>=</math>	<math> \biggl[ \biggl( \frac{\partial\lambda_2}{\partial x} \biggr)^2 + \biggl( \frac{\partial\lambda_2}{\partial y} \biggr)^2 + \biggl( \frac{\partial\lambda_2}{\partial z} \biggr)^2 \biggr]^{-1} </math>	<math>=</math>	<math> (q^2-1)^2 \biggl(\frac{\varpi z \ell}{\lambda_2} \biggr)^2 </math>
<math>h_3^2</math>	<math>=</math>	<math> \biggl[ \biggl( \frac{\partial\lambda_3}{\partial x} \biggr)^2 + \biggl( \frac{\partial\lambda_3}{\partial y} \biggr)^2 + \biggl( \frac{\partial\lambda_3}{\partial z} \biggr)^2 \biggr]^{-1} </math>	<math>=</math>	<math> \varpi^2 </math>
where,        <math>\ell \equiv (\varpi^2 + q^4 z^2)^{-1/2}</math>.

The position vector is,

<math>\vec{x}</math>

<math>=</math>

<math> \hat{i}x + \hat{j}y + \hat{k}z </math>

<math>=</math>

<math> \hat{e}_1 (h_1 \lambda_1) + \hat{e}_2 (h_2 \lambda_2) . </math>

Vector Derivatives

For orthogonal coordinate systems, the time-rate-of-change of the three unit vectors are given by the expressions,

<math> \frac{d}{dt}\hat{e}_1 </math>	<math> = </math>	<math> \hat{e}_2 A + \hat{e}_3 B </math>
<math> \frac{d}{dt}\hat{e}_2 </math>	<math> = </math>	<math> - \hat{e}_1 A + \hat{e}_3 C </math>
<math> \frac{d}{dt}\hat{e}_3 </math>	<math> = </math>	<math> - \hat{e}_1 B - \hat{e}_2 C </math>

where,

<math> A </math>	<math> \equiv </math>	<math> \frac{\dot{\lambda}_2}{h_1} \frac{\partial h_2}{\partial \lambda_1} - \frac{\dot{\lambda}_1}{h_2} \frac{\partial h_1}{\partial \lambda_2} </math>
<math> B </math>	<math> \equiv </math>	<math> \frac{\dot{\lambda}_3}{h_1} \frac{\partial h_3}{\partial \lambda_1} - \frac{\dot{\lambda}_1}{h_3} \frac{\partial h_1}{\partial \lambda_3} </math>
<math> C </math>	<math> \equiv </math>	<math> \frac{\dot{\lambda}_3}{h_2} \frac{\partial h_3}{\partial \lambda_2} - \frac{\dot{\lambda}_2}{h_3} \frac{\partial h_2}{\partial \lambda_3} </math>

See Also

• Old discussion of Integrals of Motion

 

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation